Applications of Integration
Volumes of Revolution
Many thanks to
http://mathdemos.gcsu.edu/shellmethod/gallery/gallery.html
Take this ordinary line
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5
Revolve this line around the x axis
We form a cylinder of volume
We could find the volume by finding the volume of small disc sections
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5
If we stack all these slices…
We can sum all the volumes to get the total volume
To find the volume of a cucumber…
we could slice the cucumber into discs and find the volume of each disc.
The volume of one section:
Volume of one slice =
We could model the cucumber with a mathematical curve and revolve this curve around the x axis…
Each slice would have a thickness dx and height y.
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The volume of one section:
r = y value
h = dxVolume of one slice =
Volume of cucumber…
Area of 1 slice
Thickness of slice
Take this function…
and revolve it around the x axis
We can slice it up, find the volume of each disc and sum the discs to find the volume…..
Radius = y
Area =
Thickness of slice = dx
Volume of one slice=
Divide the region into strips
Form a cylindrical slice
Repeat the procedure for each strip
To generate this solid
Regions that can be revolved using disc method
Regions that cannot….
Model this muffin.
Revolving around the x axis
Region bounded between y = 1, x = 0,
y = 1
x = 0
Volume generated between two curves
y= 1
Area of cross section..f(x)
g(x)
Your turn: Region bounded between x = 0, y = x,
Region bounded between y =1, x = 1
Region bounded betweeny = 1, x = 1
Region bounded between
Around the x axis- set it up
Revolving shapes around the y axis
Region bounded between
Volume of one washer is
Calculate the volume of one washer
And again…region bounded betweeny=sin(x), y = 0.
Region bounded between x = 0, y = 0, x = 1,
Worksheet 5Delta Exercise 16.5